Κυματική Εξίσωση
Κυματική Εξίσωσις Wave Equation thumb|300px| [[Κυματική Συνάρτηση ]] thumb|300px| [[Κυματική Συνάρτηση ]] thumb|300px| [[Επιστημονικός Νόμος Επιστημονικοί Νόμοι ---- Μαθηματικό Θεώρημα Νόμοι Μαθηματικών ---- Φυσικός Νόμος Νόμοι Φυσικής ---- Νόμοι Χημείας ---- Νόμοι Γεωλογίας ---- Νόμοι Βιολογίας ---- Νόμοι Οικονομίας ]] thumb|300px| [[Επιστήμη Επιστήμες Φυσικές Επιστήμες Βιο-Επιστήμες Γεω-Επιστήμες Οικονομικές Επιστήμες Θεωρητικές Επιστήμες Κοινωνικές Επιστήμες Επιστήμες Υγείας ---- Τεχνολογία ---- Επιστημονικός Κλάδος Επιστημονικός Νόμος Επιστημονική Μέθοδος Επιστημονική Θεωρία Επιστημονικά Κέντρα Γης Επιστήμονες Γης ]] thumb|300px| [[Διαφορική Εξίσωση Διαφορική Ανάλυση Συνήθης Διαφορική Εξίσωση Μερική Διαφορική Εξίσωση Πρωτοτάξια Διαφορική Εξίσωση Δευτεροτάξια Διαφορική Εξίσωση ]] - Νόμος της Φυσικής. - Ακριβέστερα, είναι ένας νόμος της Κυματικής - Χρονολογία ανακάλυψης. Ετυμολογία Η ονομασία "Κυματική" σχετίζεται ετυμολογικά με την λέξη "κύμα". Εισαγωγή The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable t , one or more spatial variables x_1, x_2, ...., x_n , and a scalar function u = u (x_1, x_2, ... x_n; t , whose values could model, for example, the mechanical displacement of a wave. The wave equation for u is : { \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u where ∇2 is the (spatial) Laplacian and c'' is a fixed constant. Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant ''c is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle. The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The wave equation, and modifications of it, are also found in elasticity, quantum mechanics, plasma physics and general relativity. Scalar wave equation in one space dimension The wave equation in one space dimension can be written as follows: : { \partial^2 u \over \partial t^2 } = c^2 { \partial^2 u \over \partial x^2 } . This equation is typically described as having only one space dimension "x", because the only other independent variable is the time "t". Nevertheless, the dependent variable "u" may represent a second space dimension, if, for example, the displacement "u" takes place in y-direction, as in the case of a string that is located in the x-y plane. Derivation of the wave equation The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. Another physical setting for derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). From Hooke's law The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: Imagine an array of little weights of mass m'' interconnected with massless springs of length ''h . The springs have a spring constant of k'' Here the dependent variable ''u(x) measures the distance from the equilibrium of the mass situated at x'', so that ''u(x) essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass m'' at the location ''x+''h'' are: : F_{\mathit{Newton}}=m \cdot a(t)=m \cdot d\xi\,d\eta. \, It is apparent that the solution at (t'',''x,y'') depends not only on the data on the light cone where : (x -\xi)^2 + (y - \eta)^2 = c^2 t^2, \, but also on data that are interior to that cone. Scalar wave equation in general dimension and Kirchhoff's formulae We want to find solutions to ''utt−Δ''u'' = 0 for u'' : '''Rn'' × (0, ∞) → '''R with u''(''x, 0) = g''(''x) and ut(x'', 0) = ''h(x''). See Evans for more details. Odd dimensions Assume ''n ≥ 3 is an odd integer and g'' ∈ ''Cm''+1('R''n''), h'' ∈ ''Cm('''Rn'') for ''m = (n''+1)/2. Let \gamma_n = 1\cdot 3 \cdot 5 \cdot .. \cdot (n-2) and let : u(x,t) = \frac{1}{\gamma_n}\left \left (\frac{1}{t} \partial_t \right )^{\frac{n-3}{2}} \left (t^{n-2} \int^{\text{average}}_{\partial B_t(x)} g dS \right ) + \left (\frac{1}{t}\partial_t \right )^{\frac{n-3}{2}} \left (t^{n-2} \int^{\text{average}}_{\partial B_t(x)} h dS \right ) \right then :''u ∈ C''2('R''n'' × [0, ∞)) :utt−Δ''u'' = 0 in '''Rn'' × (0, ∞) : \begin{align} \lim_{(x,t)\to (x^0,0)} u(x,t) &= g(x^0) \\ \lim_{(x,t)\to (x^0,0)} u_t(x,t) &= h(x^0) \end{align} Even dimensions Assume ''n ≥ 2 is an even integer and g'' ∈ ''Cm''+1('R''n''), h'' ∈ ''Cm('''Rn''), for ''m = (n''+2)/2. Let \gamma_n = 2 \cdot 4 \cdot .. \cdot n and let : u(x,t) = \frac{1}{\gamma_n} \left \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \int^{\text{average}}_{B_t(x)} \frac{g}{(t^2 - |y - x|^2)^{\frac{1}{2}}} dy \right ) + \left (\frac{1}{t} \partial_t \right )^{\frac{n-2}{2}} \left (t^n \int^{\text{average}}_{B_t(x)} \frac{h}{(t^2 - |y-x|^2)^{\frac{1}{2}}} dy \right ) \right then :''u ∈ C''2('R''n'' × [0, ∞)) :utt−Δ''u'' = 0 in '''Rn'' × (0, ∞) : \begin{align} \lim_{(x,t)\to (x^0,0)} u(x,t) &= g(x^0)\\ \lim_{(x,t)\to (x^0,0)} u_t(x,t) &= h(x^0) \end{align} Problems with boundaries One space dimension The Sturm-Liouville formulation A flexible string that is stretched between two points ''x = 0 and x'' = ''L satisfies the wave equation for t'' > 0 and 0 < ''x < L''. On the boundary points, ''u may satisfy a variety of boundary conditions. A general form that is appropriate for applications is : -u_x(t,0) + a u(t,0) = 0, \, : u_x(t,L) + b u(t,L) = 0,\, where a'' and ''b are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective a'' or ''b approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form : u(t,x) = T(t) v(x).\, A consequence is that : \frac{T''}{c^2T} = \frac{v''}{v} = -\lambda. \, The eigenvalue λ must be determined so that there is a non-trivial solution of the boundary-value problem : v'' + \lambda v=0, \, : -v'(0) + a v(0) = 0, \quad v'(L) + b v(L)=0.\, This is a special case of the general problem of Sturm–Liouville theory. If a'' and ''b are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for u'' and ''ut can be obtained from expansion of these functions in the appropriate trigonometric series. Investigation by numerical methods Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model: If each mass point has the mass m'', the tension of the string is ''f, the separation between the mass points is Δ''x'' and ui, i'' = 1, ..., ''n are the offset of these n'' points from their equilibrium points (i.e. their position on a straight line between the two attachment points of the string) the vertical component of the force towards point ''i+1 is : \frac{u_{i+1}-u_i}{\Delta x}\ f\,\,\,\,\,\, (1) and the vertical component of the force towards point i''−1 is : \frac{u_{i-1}-u_i}{\Delta x}\ f \,\,\,\, (2) Taking the sum of these two forces and dividing with the mass ''m one gets for the vertical motion: : \ddot u_i=\left(\frac{f}{m\ \Delta x} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right)\,\,\,\, (3) As the mass density is : \rho = \frac{m}{\Delta x} this can be written : \ddot u_i=\left(\frac{f}{\rho\ {\Delta x}^2} \right) \left(u_{i+1} + u_{i-1}\ -\ 2u_i\right) \,\,\,\,\, (4) The wave equation is obtained by letting Δ''x'' → 0 in which case ui(t'') takes the form ''u(x'', ''t) where u''(''x, t'') is continuous function of two variables, \ddot u_i takes the form \partial^2 u \over \partial t^2 and : \frac{u_{i+1} + u_{i-1}\ -\ 2u_i} If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations (Equation (5), (Equation (6) and (Equation (7)) or equivalently 200 coupled first order differential equations. Propagating these up to the times : \frac{L}{c}\ k\ 0.05\ \ k=0,\cdots ,5 The red curve is the initial state at time zero at which the string is "let free" in a predefined shape The initial state for "Investigation by numerical methods" is set with quadratic splines as follows: : u(0,x)= u_0\ \left(1-\left(\frac{x-x_1}{x_1}\right)^2\right) for 0 \le x \le x_2 : u(0,x)= u_0\ \left({\frac{x-x_3}{x_1}}\right)^2 for x_2 \le x \le x_3 : u(0,x)= 0 for x_3 \le x \le L with x_1= \frac{1}{10}\ L\ ,\ x_2=x_1+\sqrt{\frac{1}{2}}\ x_1\ ,\ x_3=x_2+\sqrt{\frac{1}{2}}\ x_1 with all \dot u_i=0 . The blue curve is the state at time \frac{L}{c}\ 0.25 , i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity c=\sqrt{\frac{f}{\rho}} would need for one fourth of the length of the string. Figure 3 displays the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=6,\cdots ,11 . The wave travels in direction right with the speed c=\sqrt{\frac{f}{\rho}} without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form f''(''x−''ct''). Figure 4 displays the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=12,\cdots ,17 . The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string. Figure 5 displays the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,23 when the direction of motion is reversed. The red, green and blue curves are the states at the times \frac{L}{c}\ k\ 0.05\ \ k=18,\cdots ,20 while the 3 black curves correspond to the states at times \frac{L}{c}\ k\ 0.05\ \ k=21,\cdots ,23 with the wave starting to move back towards left. Figure 6 and figure 7 finally display the shape of the string at the times \frac{L}{c}\ k\ 0.05\ \ k=24,\cdots ,29 and \frac{L}{c}\ k\ 0.05\ \ k=30,\cdots ,35 . The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6 Several space dimensions The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D'' in ''m-dimensional x'' space, with boundary ''B. Then the wave equation is to be satisfied if x'' is in ''D and t'' > 0. On the boundary of ''D, the solution u'' shall satisfy : \frac{\part u}{\part n} + a u =0, \, where ''n is the unit outward normal to B'', and ''a is a non-negative function defined on B''. The case where ''u vanishes on B'' is a limiting case for ''a approaching infinity. The initial conditions are : u(0,x) = f(x), \quad u_t(0,x)=g(x), \, where f'' and ''g are defined in D''. This problem may be solved by expanding ''f and g'' in the eigenfunctions of the Laplacian in ''D, which satisfy the boundary conditions. Thus the eigenfunction v'' satisfies : \nabla \cdot \nabla v + \lambda v = 0, \, in ''D, and : \frac{\part v}{\part n} + a v =0, \, on B''. In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary ''B. If B'' is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation. If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order. Inhomogeneous wave equation in one dimension The inhomogeneous wave equation in one dimension is the following: : c^2 u_{x x}(x,t) - u_{t t}(x,t) = s(x,t) \, with initial conditions given by : u(x,0)=f(x) \, : u_t(x,0)=g(x) \, The function ''s(x'', ''t) is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism. One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point (xi, ti), the value of u''(''xi, ti) depends only on the values of f''(''xi+''cti'') and f''(''xi−''cti'') and the values of the function g''(''x) between (xi−''cti'') and (xi+''cti''). This can be seen in d'Alembert's formula, stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is c'', then no part of the wave that can't propagate to a given point by a given time can affect the amplitude at the same point and time. In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point (''xi, ti) as RC. Suppose we integrate the inhomogeneous wave equation over this region. : \iint \limits_{R_C} \left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) \right ) dx dt = \iint \limits_{R_C} s(x,t) dx dt. To simplify this greatly, we can use Green's theorem to simplify the left side to get the following: : \int_{ L_0 + L_1 + L_2 } \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) = \iint \limits_{R_C} s(x,t) dx dt. The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute : \int^{x_i + c t_i}_{x_i - c t_i} - u_t(x,0) dx = - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx. In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus dt = 0. For the other two sides of the region, it is worth noting that x''±''ct is a constant, namingly xi±''cti'', where the sign is chosen appropriately. Using this, we can get the relation dx±''cdt'' = 0, again choosing the right sign: : \begin{align} \int_{L_1} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) &= \int_{L_1} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right)\\ &= c \int_{L_1} d u(x,t) \\ &= c u(x_i,t_i) - c f(x_i + c t_i). \end{align} And similarly for the final boundary segment: : \begin{align} \int_{L_2} \left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \right ) &= - \int_{L_2} \left ( c u_x(x,t) dx + c u_t(x,t) dt \right )\\ &= - c \int_{L_2} d u(x,t) \\ &= c u(x_i,t_i) - c f(x_i - c t_i). \end{align} Adding the three results together and putting them back in the original integral: : \begin{align} \iint_{R_C} s(x,t) dx dt &= - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) \\ &= 2 c u(x_i,t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) dx \end{align} Solving for u''(''xi, ti) we arrive at : u(x_i,t_i) = \frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} + \frac{1}{2 c}\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + \frac{1}{2 c}\int^{t_i}_0 \int^{x_i + c \left ( t_i - t \right )}_{x_i - c \left ( t_i - t \right )} s(x,t) dx dt. In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (xi, ti) compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source. Other coordinate systems In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation. Further generalizations Elastic waves The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion: : \rho{ \ddot{\bold{u}}} = \bold{f} + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold{u}) - \mu\nabla \times (\nabla \times \bold{u}) where: *λ and μ are the so-called Lamé parameters describing the elastic properties of the medium, *ρ is the density, *'f' is the source function (driving force), *and u''' is the displacement vector. Note that in this equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if '''f and ∇⋅'u' are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field E', which has only transverse waves. Nonlinear dispersion relation In dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation : \omega=\omega(\bold{k}), where ''ω is the angular frequency and '''k is the wavevector describing plane wave solutions. For light waves, the dispersion relation is ω'' = ±''c |'k'|, but in general, the constant speed c'' gets replaced by a variable phase velocity: : v_\mathrm{p} = \frac{\omega(k)}{k}. Ομογενής Κυματική Εξίσωση Η ομογενές εξίσωση κύματος σε μία διάσταση : \frac 1{c^2}\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2} =0 έχει τη γενική λύση: : u\left(t, x\right) = f(x + ct) + g(x - ct) :με αυθαίρετες διαφορίσιμες συναρτήσεις f(x)\, και g(x)\, . :όπου: *ο πρώτος όρος περιγράφει κύμα που διαδίδεται προς τα αριστερά με ταχύτητα c και *ο δεύτερος όρος περιγράφει κύμα που διαδίδεται προς τα δεξιά με την ίδια ταχύτητα © *Οι συναρτήσεις f(x)\, και g(x)\, ονομάζονται Riemann-Invariants. Αντί των συναρτήσεων f και g μπορεί να χρησιμοποιηθεί ο γραμμικός συνδυασμός *με μορφή συνημιτόνου: : \cos(k x - \omega t + \varphi) *ή με μορφή εκθετικού : \mathrm{e}^{\mathrm{i}(k x - \omega t)}\, Η λύση μπορεί να αποδοθεί από το πραγματικό τμήμα ενός ολοκληρώματος: : u(t,x)=\text{Re}\int\mathrm d k\,a(k)\, \mathrm{e}^{\mathrm{i}(k\, x -\omega\,t)} Η σχέση συχνότητας και κυματικού αριθμού είναι: : \omega = |k|\,c Σημειωτέον ότι η φάση \varphi{(k)} έχει ενσωματωθεί στο πλάτος a(k) . Υποσημειώσεις Εσωτερική Αρθρογραφία * Δονούμενη Χορδή *Acoustic attenuation *Acoustic wave equation *Bateman transform *Electromagnetic wave equation *Helmholtz equation *Inhomogeneous electromagnetic wave equation *Laplace operator *Mathematics of oscillation *Maxwell's equations *Schrödinger equation *Standing wave *Vibrations of a circular membrane *Wheeler–Feynman absorber theory * Φυσικός Νόμος * Εξίσωση *Διαφορική Εξίσωση *Μαθηματική Συνθήκη *Κύμα *Κυματική *Δονούμενη Χορδή Βιβλιογραφία * M. F. Atiyah, R. Bott, L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients I", ''Acta Math., 124 (1970), 109–189. * M.F. Atiyah, R. Bott, and L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients II", Acta Math., 131 (1973), 145–206. * R. Courant, D. Hilbert, Methods of Mathematical Physics, vol II. Interscience (Wiley) New York, 1962. * L. Evans, "Partial Differential Equations". American Mathematical Society Providence, 1998. * "Linear Wave Equations", EqWorld: The World of Mathematical Equations. * "Nonlinear Wave Equations", EqWorld: The World of Mathematical Equations. * William C. Lane, "MISN-0-201 The Wave Equation and Its Solutions", Project PHYSNET. Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia * Nonlinear Wave Equations by Stephen Wolfram and Rob Knapp, Nonlinear Wave Equation Explorer by Stephen Wolfram, and Wolfram Demonstrations Project. * Mathematical aspects of wave equations are discussed on the Dispersive PDE Wiki. * Graham W Griffiths and William E. Schiesser (2009). Linear and nonlinear waves. Scholarpedia, 4(7):4308. doi:10.4249/scholarpedia.4308 * πό την εξίσωση του αρμονικού κύματος στην εξίσωση Dirac * Κυματική Εξίσωση, Ευταξίας Category: Διαφορικές Εξισώσεις Category: Νόμοι Κυματικής